In 1968, J. Simons proved a very important formula for the Laplacian of the second fundamental form of a minimal submanifold in a Riemannian manifold and then used it to characterize certain minimal submanifolds of a sphere and Euclidean space. Over the years, such formulas, called Simons type equations, also proved to be a powerful tool for studying cmc and pmc submanifolds.

In this talk, we shall present two Simons type equations for submanifolds with parallel mean curvature (pmc submanifolds) in \(M^n(c)\times\mathbb{R}\), where \(M^{n}(c)\) is an \(n\)-dimensional space form with constant sectional curvature \(c\), and then some applications for pmc surfaces in \(M^3(c)\times\mathbb{R}\), and for biharmonic pmc submanifolds of any codimension.

The talk is based on three papers, which are joint works with Harold Rosenberg (IMPA, Rio de Janeiro, Brazil), one of them being also co-authored by Cezar Oniciuc (UAIC, Iași, Romania).

Metamathematics is the study of mathematical methods: what is possible, or impossible, provable or unprovable, or undecidable using given methods. Old examples of metamathematical results include ruler-and-compass impossibility theorems, non-Euclidean geometries, the Abel-Ruffini theorem. I will first sketch the history of metamathematical results and ideas and then explain where we stand now: how to prove that something is unprovable and show some examples of interesting unprovable statements. I will especially concentrate on unprovability in concrete finite combinatorics. Then I will show some of the latest theorems by myself, and my collaborators on unprovability in combinatorics and state the biggest problems facing the subject. This will be a talk accessible to all mathematicians.